Transcript EFTC10 Oral Presentation

Profile-turbulence interactions, MHD relaxations and
transport in Tokamaks
A Thyagaraja*, P.J. Knight*,
M.R. de Baar†, G.M.D. Hogeweij† and E.Min†
*UKAEA/EURATOM Fusion Association
Culham Science Centre, Abingdon, OX14 3DB, UK
†Assoc. EURATOM-FOM, Trilateral Euregio Cluster, P.O. Box 1207, 3430
BE Nieuwegein, The Netherlands
IAEA Meeting, Trieste, Mar 2-4, 2005
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Acknowledgements
• Jack Connor, Jim Hastie, Chris Gimblett, Martin Valovič,
Ken McClements, Terry Martin, Chris Lashmore-Davies
(Culham)
•
Niek Lopes Cardozo (FOM)
• Xavier Garbet, Paola Mantica, Luca Garzotti (EFDA/JET)
• EPSRC (UK)/EURATOM
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Synopsis
• Role of profile-turbulence interactions and spectral transfer
processes in tokamak turbulence and transport
• The key concepts: spectral cascades, profile-turbulence
interactions, nonlinear self-organization, dynamos, zonal
flows
• Some typical simulation results from CUTIE and
comparisons with experiment
• Conclusions
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Characteristics of tokamak “plasma climatology”
• Universal, electromagnetic turbulence, between system
size and ion gyro radius; confinement (s) and Alfvén (ns)
times.
• Strong interactions between large and small scales;
inhomogeneity of turbulence.
• Plasma is strongly “self-organising”, like planetary
atmospheres (Rossby waves=Drift waves).
• Transport barriers connected with sheared flows, rational
q’s, inverse cascades/modulational instabilities
(Hasegawa).
• Analogous to El Nino, circumpolar vortex, “shear
sheltering” (J.C.R Hunt et al):
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Profile-turbulence interactions
• All plasma instability, linear or nonlinear, caused by thermal
disequilibrium in a driven-dissipative system
• Profile-turbulence cross-talk: turbulence corrugates profiles; latter
saturate turbulence. Both electrostatic and magnetic components
interact strongly and play a role
• Macroscale phenomena (pellets, sawteeth, ELM’s, ITB’s,..)
influence and are influenced by mesoscale turbulence (possibly
also micro scale): nonlinear self-organization
• Momentum/angular momentum exchanges between turbulence
and “mean profiles” result in dynamo currents (electrons) and
zonal flows (ions).
• No real “scale separation”-a continuum of scales in time and
space
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Spectral Transfer Mechanisms
Nonlinearity; phase mixing by flows & Alfven waves
Direct cascade
ExB;jxB
Zonal
flows
Random
phases
Streamers
Turbulent
diffusion
Dynamo
currents
Macroscale
Inverse cascade
Mesoscale
Microscale
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Modulational Instabilities; beating
“Arithmetizing” two-fluid plasma turbulence:CUTIE
• Global, electromagnetic, two-fluid code.Co-evolves turbulence
and equilibrium-”self-consistent” transport.
• “Minimalist plasma climatology” : Conservation Laws and
Maxwell’s equations for 7-fields, 3-d, pseudo spectral+radial
finite-differencing, semi-implicit predictor-corrector, fully nonlinear.
• Periodic cylinder model, but field-line curvature treated; describes
mesoscale, fluid-like instabilities; no kinetics or trapped particles
(but includes neoclassics).
• Very simple sources/boundary conditions (overly simple
perhaps?!)
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Off-axis ECH in RTP
[Phys Rev Letts.- de Baar et al, 94, 035002, (2005)]
• Ip=80 kA, Bf=2.24 T, qa=5.0, Hydrogen plasma
• neav ~ 3.0 E+19 m-3 PECH~350 kW, P ~80 kW
• PECH deposited at r/a = 0.55
• Resolution: 100x32x16; dt=25 ns ; simulated for >50 ms
   (%)  0.16;  *  0.007; *  1.0
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Initial and Averaged Profiles:Te,Ti,ne,q
(Squares-experiment; solid line-CUTIE)
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Power density and Electron advective
Heat flux Profiles
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Time-averaged Zonal Flow (-cEr/B) and
Current density components
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Zonal Flows
• Poloidal E x B flows, turbulent Reynolds stresses:
“Benjamin-Feir” type of modulational instability, “inverse
cascade” recently explained in Generalized Charney
Hasegawa Mima Equation
• McCarthy et al. PRL, 93, 065004, 2004
• Highly sheared transverse flows “phase mix” and lead to a
“direct cascade” in the turbulent fluctuations.
• Enhances diffusive damping and stabilizes turbulence
linearly and nonlinearly. Confines turbulence to low shear
zones.
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Zonal Flow Evolution
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Current/q Profile Evolution
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Barriers and q
• CUTIE naturally tends to produce barriers near the simple
rationals in q.(only global codes can do this!)
• Mechanism: heating > mode> asymmetric turbulent
fluxes> zonal flow and dynamo effects> reduce high-k
turbulence and flatten q>local reduction of advection
• >higher pressure gradients>relaxation oscillation
• Two barrier loops operate in CUTIE! The loops interact in
synergy.
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Outbound heat flow and "ears"
• Off-axis ECH-power enhances the MHD level near the
deposition radius.
• The interplay of the EM-and ES-component of these
fluctuations gives rise to an outward heat-flow.
• This is sufficient for supporting pronounced off-axis Te
maxima in CUTIE, comparable with expt.
• The ears are quite comparable to the experimental
observations.
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Barriers and q
Off-axis Sawteeth simulated by CUTIE:
Te, q at r/a=0, 0.55
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“Ear choppers”: CUTIE vs. Expt.
CUTIE
RTP
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Sawtooth details and Magnetic and Electrostatic
turbulence evolution in CUTIE
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Off-axis sawteeth: comparison with RTP
• CUTIE produces MHD events (as in experiment) associated
with profile-turbulence interactions, zonal "jets", "elbows" in
the q profile; relaxations called “ear choppers”.
•
•
•
•
CUTIE Period (~3 ms),
CUTIE Amplitude (~150-200 eV)
CUTIE Crash time (~0.3 ms)
CUTIE Conf. time (~3-4 ms)
RTP (~1.5-2 ms)
RTP (~100 eV)
RTP (~0.2-0.5 ms)
RTP (~3 ms)
• “Avalanching” and “bursts”; intermittency outside heating
radius.
• Qualitative agreement with experiment.
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No dynamo, no sawteeth!
With dynamo
No dynamo
Volume averaged magnetic turbulence measure and loop voltage
No "precursors" but "postcursors" in magnetic turbulence
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High resolution study of Ohmic sawteeth
[& ELM’s ?!]
• Ip=90 kA, Bf=2.24 T, qa=5.0, Hydrogen plasma
• neav ~ 3.0 E+19 m-3 P ~90 kW; Zeff= 2-4; Edge source
• Resolution: 100x64x16; dt=25 ns ; simulated for >25 ms
• Movies of profiles: ne, Te, V(zonal)= -cEr/B, j(dynamo), j(bs)
• Contours: Te, radial ExB, A-parallel fluctuations
   (%)  0.14;  *  0.0067;*  0.6
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Ohmic m=1 sawteeth & edge instability: V-loop, Beta
Te(0)~800 eV (CUTIE) close to RTP~760 eV; monotonic
ne(0) 4.0 E+19 (CUTIE) RTP 5.0 E+19
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Ohmic RTP case:averaged Te,Ti,ne,q
(Squares-experiment; solid line-CUTIE)
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Movie!
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Question: What does this model predict?
• Do CUTIE results bear a qualitative resemblance to experiments
(RTP, MAST, JET, FTU,..)? (Conditional “yes”!)
• Is there any quantitative agreement? (in some cases and fields)
• What have we learned from CUTIE simulations? (profileturbulence interaction crucial)
• What are the limitations of minimalism and how can one proceed
further? (many effects omitted; do they matter? Occam’s Razor!)
• What are the lessons (if any) for the future? (go from “large” to
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“small” scale)
Conclusions-I
• “Minimalist CUTIE model” applied to RTP, JET, MAST, FTU,
TEXTOR, T-10
•
First "turbulence code" to describe ”on and off-axis sawteeth"
dynamically in experimental conditions
•
Describes self-organization caused by profile-turbulence
interactions
•
Insight into spectral transfer & spontaneously generated
zonal flows and dynamo currents in tokamaks
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Conclusions-II
• Illuminates role of turbulence in shaping large-scale behaviour
& demonstrates features of experiment:
1) key role of rational q surfaces and electromagnetic modes
2) off-axis maxima and outward heat advection (“ears”)
3) role played by “corrugated” zonal flows, MHD relaxation
4) deep and shallow pellet behaviour in JET(with ITB's)
• Complementary to gyrokinetics: better suited to long-term
evolutionary studies (“plasma climatology”) and global,
electromagnetic, meso plasma dynamics.
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Discussion
• CUTIE's "minimalist" model used globally, provides synoptic
description of a range of dynamic phenomena involving
turbulence and transport: MECH, pellets, MHD relaxation, ITB’s
• Limitations/ short-comings:
• Geometry
• Trapped particle physics, kinetic effects
• Atomic physics effects, radiation, impurities
• Proper source terms
• ”Real time" (ie fast!) calculations and effective predictions to guide
experiments, diagnostics and design.
• Higher resolution in space (with correct physics!)
• Worries about missing "microscale” physics. (Is the Earth’s climate
influenced by air turbulence on a 10x10x10 m grid?)
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Spectral transfer mechanisms
• Electromagnetic turbulence due to linear/nonlinear instability:
spontaneous symmetry breaking-results in spectral cascades
(both direct and inverse).
• Sheared flows and Alfven waves cascade (particularly enstrophy)
to high radial k. Landau damping/phase-mixing “kills” fine-scale
structures (if they exist, “where are they?”)
• Two high-k linearly growing modes can “beat” to populate the lowk and can also decay strongly by modulational instability: a
fundamental “inverse spectral cascade” (Hasegawa, LashmoreDavies et al, Benjamin-Feir)
• Powerful means to “self-generate” equilibrium flows & currents
and populate low-k spectrum forming “condensates”
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Generic Transport Equation & Flux
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Equations of Motion (in brief!)
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Equations of Motion (2)
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Two barrier loops in CUTIE
Asymmetric fluxes near mode
rational surface
Driving terms
of turbulence
Pressure gradient
Zonal flows modify
turbulence-back reacts
Turbulent dynamo,
currents
q, dq/dr, j, dj/dr
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The Advection-Diffusion Equation
f

t
f

f
v y (x) y  x ( D x )
Sheared velocity in combination with diffusion changes spectrum
R  (v' ( x ) Lx ) / D
2
“Reynolds number” measures shear/diffusion:
v'  D
2/3
Damping rate is proportional to
y
y
1/ 3

v' R
1 / 3
y
Spectrum discrete, “direct cascade due to phase mixing”
“Jets” in velocity lead to “ghetto-isation/confinement” to low shear
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regions
Zonal Flow (-cEr/B) Evolution:
corrugations
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Total current density and dynamo current density
evolution
Current is expelled from core and strong profile flattening
Corrugated dynamo current (both signs!); localization
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RTP tokamak: well-diagnosed, revealing subtle features of
transport, excellent testing ground
Step-like
changes
in Te(0)
“plateaux”
whenever
deposition
radius
crosses
“rational”
surfaces!
Sawtooth like oscillations
A
A’
Te(0)
A”
B
C
D
E
Hollow Te
ECH power deposition radius (Rho/a)
0.5
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RTP Experimental Te profiles for different ECH
deposition radii
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Zonal flow (-cEr/B) and bootstrap current density
Negative values of zonal flow indicate ion diamagnetic flow values; note
corrugations in both fields (j-bs is typically positive)
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Equations solved: reduced forms
Continuity
Energy
Parallel
momentum
Potential vorticity
Quasi-neutrality
Ohm+Faraday
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